We present our progress on the computation of the axial form factor of the nucleon with flavour structure $u+d-2s$ from lattice QCD. We employ a set of $N_f=2+1$ CLS ensembles with $O(a)$-improved Wilson fermions and the Lüscher-Weisz gauge action, with lattice spacings ranging from
$0.05\,\text{fm}$ to $0.086\,\text{fm}$ and pion masses spanning between $130\,\text{MeV}$ and $350\,\text{MeV}$.
We employ multiple source-sink separations and use the summation method to suppress the contamination from excited states.
We use a $z$-expansion on each ensemble to parametrize the $Q^2$-dependence of the form factor and simultaneously fit the available source-sink separations for all
$Q^2\leq 0.7 \,{\rm GeV}^2$.
We outline our analysis of the stability of the fits varying the ansätze and different estimations of the covariance matrix and report on our strategy for a comprehensive determination of the physical form factor.